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A201998 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c such that n = c*x^2 + (c + 1)*x + c*41 for an integer x. 4
244, 249, 251, 266, 270, 295, 301, 336, 344, 389, 399, 407, 416, 418, 445, 449, 454, 466, 489, 494, 496, 500, 506, 527, 531, 545, 547, 563, 570, 571, 582, 583, 585, 611, 612, 620, 622, 624, 628, 630, 636, 652, 661, 662, 663, 679, 693, 699 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The composition of functions k(x) factors. k(x) = (x^2 + x + 41)*(c^2*x^2 + (c^2 + 2*c)*x + c^2*41 + c + 1).  So k(x) is the product of two integers greater than one and thus composite.

REFERENCES

John Stillwell, Elements of Number Theory, Springer, 2003, page 3.

LINKS

Table of n, a(n) for n=1..48.

Matt C. Anderson A prime producing polynomial writeup

MAPLE

maxn:=1000:

A:={}:

for n from 1 to maxn do

g:=n^2+n+41:

if isprime(g)=false then

A:=A union {n}:

end if:

end do:

# The set A contains values n such that n^2+n+41 is composite and n < maxn.

c:=1:

x:=-1:

p:=41:

q:=c*x^2-(c+1)*x+c*p:

A2:=A:

while q < maxn do

while q < maxn do

A2:=A2 minus {q}:

A2:=A2 minus {c*x^2+(c+1)*x+c*p}:

x:=x+1:

q:=c*x^2-(c+1)*x+c*p:

end do:

c:=c+1:

x:=-1:

q:=c*x^2-(c+1)*x+c*p:

end do:

A2;

MATHEMATICA

Reap[For[n=1, n<700, n++, If[!PrimeQ[n^2+n+41], If[Reduce[c>0 && n == c*x^2+(c+1)*x+41*c , {c, x}, Integers] === False, Sow[n]]]]][[2, 1]] (* Jean-Fran├žois Alcover, Apr 30 2014 *)

CROSSREFS

Cf. A007634 (n^2 + n + 41 is composite).

Cf. A235381 (similar to this sequence).

Sequence in context: A243774 A051002 A044987 * A234262 A031786 A328206

Adjacent sequences:  A201995 A201996 A201997 * A201999 A202000 A202001

KEYWORD

nonn

AUTHOR

Matt C. Anderson, Dec 07 2011

STATUS

approved

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Last modified August 11 03:41 EDT 2022. Contains 356046 sequences. (Running on oeis4.)